逆矩阵的几个属性:
det(A)
!= 0消元法 = 因式分解: $A = LU$
A = [1 3 1
1 1 -1
3 11 6]
3×3 Array{Int64,2}: 1 3 1 1 1 -1 3 11 6
iA = inv(A)
3×3 Array{Float64,2}: -8.5 3.5 2.0 4.5 -1.5 -1.0 -4.0 1.0 1.0
A * iA
3×3 Array{Float64,2}: 1.0 2.22045e-16 1.11022e-16 0.0 1.0 -1.11022e-16 0.0 -3.55271e-15 1.0
det(A * iA)
1.0000000000000009
L, U = lu(A, Val{false})
L
3×3 Array{Float64,2}: 1.0 0.0 0.0 1.0 1.0 0.0 3.0 -1.0 1.0
U
3×3 Array{Float64,2}: 1.0 3.0 1.0 0.0 -2.0 -2.0 0.0 0.0 1.0
L * U
3×3 Array{Float64,2}: 1.0 3.0 1.0 1.0 1.0 -1.0 3.0 11.0 6.0
[1 2 3; 4 5 6]'
3×2 Array{Int64,2}: 1 4 2 5 3 6
Q = [1/3 2/3 2/3; 2/3 1/3 -2/3; 2/3 -2/3 1/3]
isapprox(inv(Q), Q')
true
置换矩阵就是每一行每一列都有一个 1. 记为 $P$, 而 $P^{T}$ 依然是置换矩阵。单位矩阵 $I$ 是最简单的置换矩阵。通过交换 $I$ 的行,可以得到全部可能的置换矩阵。A permutation matrix $P$ has the rows of the identity $I$ in any order.
本质上,置换矩阵是对行交换的顺序的一种描述。 $PA$ 的含义就是,对 $A$ 进行 $P$ 所描述的行交换。
B = [4 -2 -7 -4 -8
9 -6 -6 -1 -5
-2 -9 3 -5 2
9 7 -9 5 -8
-1 6 -3 9 6]
5×5 Array{Int64,2}: 4 -2 -7 -4 -8 9 -6 -6 -1 -5 -2 -9 3 -5 2 9 7 -9 5 -8 -1 6 -3 9 6
L, U, p = lu(B)
L
5×5 Array{Float64,2}: 1.0 0.0 0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.444444 0.0512821 1.0 0.0 0.0 -0.111111 0.410256 0.582822 1.0 0.0 -0.222222 -0.794872 0.171779 0.0242696 1.0
U
5×5 Array{Float64,2}: 9.0 -6.0 -6.0 -1.0 -5.0 0.0 13.0 -3.0 6.0 -3.0 0.0 0.0 -4.17949 -3.86325 -5.62393 0.0 0.0 0.0 8.67894 9.95297 0.0 0.0 0.0 0.0 -0.771206
p
5-element Array{Int64,1}: 2 4 1 5 3